Question

What is the graph of `y = i^x` ?

Answer 1

Considered as a function from real numbers `RR` to complex numbers `CC`, the graph of this function is a helix in three dimensions.

Explanation:

Considered as a function from `RR` to `CC`, as `x` moves along the `x`-axis, `y` will move round the unit circle in the complex plane.

If we represent this in `3` dimensions, the graph forms a helix

Note that:

`i^x = (e^(pi/2 i))^x = e^((pi/2 x) i) = cos(pi/2 x) + i sin (pi/2 x)`

This will trace a complete circle once every time `x` increases (or decreases) by `4`